Countermeasures against High-Order Fault-Injection Attacks on CRT-RSA

In this paper we study the existing CRT-RSA countermeasures against fault-injection attacks. In an attempt to classify them we get to achieve deep understanding of how they work. We show that the many countermeasures that we study (and their variations) actually share a number of common features, but optimize them in different ways. We also show that there is no conceptual distinction between test-based and infective countermeasures and how either one can be transformed into the other. Furthermore, we show that faults on the code (skipping instructions) can be captured by considering only faults on the data. These intermediate results allow us to improve the state of the art in several ways: (a) we fix an existing and that was known to be broken countermeasure (namely the one from Shamir), (b) we drastically optimize an existing countermeasure (namely the one from Vigilant) which we reduce to 3 tests instead of 9 in its original version, and prove that it resists not only one fault but also an arbitrary number of randomizing faults, (c) we also show how to upgrade countermeasures to resist any given number of faults: given a correct first-order countermeasure, we present a way to design a provable high-order countermeasure (for a well-defined and reasonable fault model). Finally, we pave the way for a generic approach against fault attacks for any modular arithmetic computations, and thus for the automatic insertion of countermeasures.